Fig. 1. Schematic of the experimental setup. An optically smooth water thin film was generated from a gas-dynamic liquid nozzle. The thin film was heated by the FLASH XUV-FEL focused by an ellipsoidal mirror to 45 × 24 µm² FWHM. The transmitted ratio of the FEL energy was measured by a YAG screen. The optical reflection and transmission of the thin film sample were probed by 750 and 850 nm, 100 fs FWHM laser pulses generated from an optical parametric amplifier. The measurement was carried out continuously at 10 Hz. Examples of reflection and transmission data measured by the 850 nm probe at an energy density of 9.1 ± 1.2 MJ/kg are shown.

Fig. 2. Examples of the reflection and transmission data analysis. The data were acquired using the 850 nm probe at 300 fs after FEL excitation. (a) The raw reflection image on the left contains curved interference fringes, and the FEL-heated region is marked by the blue box. The average count at the center of the FEL-heated area (small red dot) is used as IRFEL in Eq. (1). The fringes inside the yellow box were flattened by a cross-correlation function, and the result is shown in the middle. The lineout taken from the flattened image is shown on the right, where IRmax and IRmin are obtained for use in Eq. (1). The gray band in the flattened image containing the FEL-heated region is excluded when we calculate the lineout curve. (b) The center of the FEL-heated region (small red dot) in the raw transmission image is used to determined ITFEL in Eq. (2). The vacuum areas in boxes B1 and B3 are used to determine the reference vacuum transmission ITv in the same equation. The distances from B1 and B3 to the FEL-heated spot are the same.

Fig. 3. Measured reflection and transmission data for a (300 ± 30) nm thin water film heated to various energy densities and measured by a 750 nm probe [(a) and (b)] and a (200 ± 20) nm thin film measured by a 850 nm probe [(c) and (d)]. The initial reflectivity in the 850 nm data is significantly higher than in the 750 nm data. This is because the 850 nm measurements were carried out near the peak intensity of the interference fringes, while the 750 nm measurements were near the minimum intensity of the fringes.

Fig. 4. Reflection and transmission measured using a probe wavelength of 750 nm, and the deduced complex refractive index and optical conductivity of water (300 nm thick) at an absorbed energy density of (6.1 ± 1) MJ/kg. (a) and (b) Averaged reflection and transmission at different time delays. (c) and (d) Complex refractive indices determined from Maxwell’s equations implemented by the transform matrix method. (e) and (f) Real and imaginary parts of the corresponding electrical conductivity. The vertical dashed lines indicate the time window of 0.2–0.6 ps, where the electrical conductivity is obtained for further discussion.

Fig. 5. Reflection and transmission measured using a probe wavelength of 850 nm, and the deduced complex refractive index and optical conductivity of water (200 nm thick) at an absorbed energy density of (6.1 ± 1) MJ/kg. (a) and (b) Averaged reflection and transmission at different time delays. (c) and (d) Complex refractive index determined from Maxwell’s equations implemented by the transform matrix method. (e) and (f) Real and imaginary parts of the corresponding electrical conductivity. The vertical dashed lines indicate a time window of 0.2–0.6 ps, where the electrical conductivity is obtained for further discussion.

Fig. 6. Real and imaginary parts of the electrical conductivity from cold and heated water (0.2–0.6 ps after FEL heating, with the error bars representing the standard deviations of the data within this time interval) on the contour plots of reflection and transmission. (a) and (b) 750 nm laser probe on a 300 nm thick sample, with the heated-state samples being measured at energy densities of (3.2 ± 0.8), (4.3 ± 0.9), (5.3 ± 0.9), (6.4 ± 1), and (8.5 ± 1.2) MJ/kg, respectively. (c) and (d) 850 nm laser probe on a 200 nm thick sample, with the heated-state samples being measured at energy densities of (3.1 ± 0.8), (4.6 ± 0.9), (6.1 ± 1), (9.1 ± 1.2), and (12 ± 1.4) MJ/kg, respectively. The room temperature data (RT, i.e., T_e = T_i = 300 K) are shown at the top left corner in each plot.

Fig. 7. (a) Electron specific heat capacity as a function of electron temperature from DFT calculations. (b) Peak electron temperature as a function of absorbed XUV energy density, calculated using Eq. (4).

Fig. 8. Electron DOS (e-DOS) of water from DFT calculations at electron temperatures of (a) 300 K and (b) 20 000 K. The ion temperature is 300 K in both cases.

Fig. 9. (a) Density of conduction (carrier) electrons as a function of T_e calculated using the results for the electron DOS and Eq. (6). (b) The corresponding electron degeneracy parameter Θ.

Fig. 10. Total structure factor of water calculated by MD simulations at equilibrium temperatures (T_e = T_i) of 300, 500, 1000, and 1500 K. The baseline of each curve is shown by the dash-dotted lines on the left using the corresponding colors. The vertical dotted line and dashed line correspond to twice the Fermi vector, i.e., the upper bound of the integral in Eq. (7) at carrier densities of 1 × 10²⁷ and 5 × 10²⁷ m⁻³, respectively.

Fig. 11. Optical conductivity as a function of electron temperature T_e: a comparison of experiments and theoretical calculations. Both DFT-MD and Ziman theory calculations were performed at T_i = 300 K. The vertical error bars are inherited from Fig. 6, and the horizontal error bars correspond to the range of energy densities in the data.